JOURNAL OF RESEARCH of the National Bureau of Standards Volume 82 , No. I, July- August 1977

Uniaxial Extension and Compression in Stress-Strain Relations of Rubber

Lawrence A. Wood

Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234

(June 6, 1977)

A survey of ex pe rimental data fro m the lite rature in cases whe re the de formation of a specime n is vari ed co ntinuous ly from uniaxial compress ion to te ns il e d eformation s hows th a t Young's Modulus M, defin ed as the limit of s tress to s train in the u ndeformed sta te, is independent of the direc ti on of approach to the limit. The norma li zed stress-stra in re la ti o n o f Malt in , Roth , a nd Sti ehl e r (MRS, 1956) is F/M = (L- I - L- 2) ex p A (L - L- I) where F is the s tress on the und e form ed section, L is the ex te ns ion rat io, a nd M and A a re consta nt s. Values of M a nd A a re obl<lined from the interce pt a nd s lope of a g raph of ex pe rime ntal observa ti ons of log F/(L- 1 - L - 2) aga ins t (L - L - I) inc luding obse rvations of uniaxial co mpress io n if ava ilab le. They fou nd the value of A to be about 0.38 for pure­gum vu lca ni zates of natural rubber a nd seve ra l sy nth e ti cs.

In la te r work several observers have now found that the eq uation is a lso va lid for vul ca ni zates co ntaini ng a fill er, but A is highe r, reac hing a va lue of about 1 for large amo unts of fill e r. In extreme cases A is not constant a t low de formation s. The range of appli cability in many cases now is found to ex tend from the co mpressive region wh ere L = 0.5 up to the po int of tens il e rupture or to a point whereA inc reases abruptl y beca use of c rys ta lliza tion . Taking A as a constant paramete r in the range 0.36 to 1, graphs a re presented showi ng calculat ed values of (1) F/M as a fun ction of L a nd (2) the normali zed Mooney- Rivlin plot of F/[2M(L - L - 2)] aga inst L - I. Each of the la tte r graphs has o nl y a limited region of linearity corresponding to cons tant values of the Mooney- Rivlin coeffi c ients C1 a nd C2 •

Si nce thi s region does not include the unde form ed sta te, where L = I , or any of the compress ion region, the utility of the Moo ney- Ri vlin eq uation is ex tremely limited , s ince it ca n not be used a t low elongations. The coeffi c ie nts a re dramatica ll y a lt e red for rubbers s how ing diffe re nt va lues of the MRS constant A. For rubbe rs show ing the higher values of A, the coeffici e nts a re radically a lte red and the region of approx imate linea rit y is drastica ll y reduced.

Key word s : Ex tens ion a nd co mpressio n in rubbe r; MaI1in-Roth-Stie hl er equation; modulu s of rubber; Mooney­Riviin equation; rubbe r; stress-strain relations; stress-stra in re lations in rubbe r; uni ax ia l ex tension and compres­s ion in rubbe r.

1. Introduction

The form of the stress-strain relation of rubber has been the subj ect of several previous investigations at the National Bureau of Standards [I). 1 An empirical eq uation was devel­oped by Martin, Roth, and Stiehler [2] in 1956 to describe the behavior of a typical pure-gum rubber vulcanizate afte r a given period of creep. They showed it to be valid up to 200 pe rcent elongation for the first extension of pure-gum vulcan­izates of natural rubber, styrene-butadiene rubber (SBR), butyl rubber (IIR), and chloroprene rubber (CR) over a 10-fold range of times of vulcanization and for constant times of c ree p (isochrones) from 1 to 10,000 min. Another paper [3] showed that the equation could be applied to the compression region with the same constants as in the extension region. This conclusion is espec ially significant because it means that the equation is valid at low deformations, where it is difficult to obtain good experimental data. It also means that the slope of the stress-strain curve shows no discontinuity on passi ng from uniax ia l compress ion to tension. As discussed later, this is in accordance with direct experimental observa­tions .

I figures in brac kets indicate the literature refere nces al the end of this paper.

The present pape r extends consideration to published ex­pe rimental data regarding the nature of th e s tress-strain relation at e longations above 200 percent up to the point of rupture. The discussion includes the effec ts of c rystallization when present and the effects of filler content. Here one must conside r values of the paramete r A in the Martin, Roth, and Stiehler eq uation which are larger than that utilized origi­nally.

A discussion is given of the limited regions of linearity of the Mooney-Rivlin plot and the relation of the constants C 1

and C 2, obtained from that plot, to the modulus (the limit of the ratio of stress to strain in the undeformed state).

2. Continuity of Slope of Stress-Strain Curves

Over many years there has been general agreement with the experimental observation that the slope of the stress­strain curve shows no discontinuity on passing from uniax ial compression to extension [4-8). In other words the modulus, M defined as the limit of the ratio of stress , F, to strain at L = 1 is inde pendent of the direction of approach to the limit, where L is the extension ratio.

The validity of this conclusion has been questioned by van del' Hoff and Glynn [9], who have presented data on polybu­tadiene vulcanizates. However, they did not make measure-

57

ments on the same specimens in extension and compression. Data of Blokland [10, 11] on lightly cross-linked polyure­thane networks have sometimes been quoted in support of a discontinuity. However Blokland [10] himself states that the data on compression are not very satisfactory because of friction arising from tackiness of the specimens.

Direct experimental studies involving a single spec imen in the two regions have been rare. R. H. Taylor, [12] in unpub­lished work in our laboratories in 1942, utilized the central section of a cylindrical specimen of pure-gum natural rubber in a balanced beam arrangement to pass in continuous fash­ion from compression to exte nsion. When L was varied from 0.9 to 1.1 , a plot of FL against L yielded a single straight line . Linearity of this plot indi cates that the ratio of stress calculated on the deformed cross-section to strain is a con­stant over this range of deformation [3], and that there is no discontinuity.

In more exte ns ive recent work Wolf [13] accomplished the same result by metal springs counte rbalanced by weights. As the extension ratio L increased from 0.88 to 1.18, the rat io of F, the stress on the initial section, to the s train decreased continuously , showing no evidence of discontinuity at L = 1.

3 . Form of Stress-Strain Relation

Three typical curves in figure 1 show the relation of stress F (based on undeformed cross-sectional area) to the extension ratio L , the ratio of stressed length to unstressed length . The figure includes a short section of the compression region where L < 1. The values of Young's modulus M , the slope of each curve for the unstressed state (where L = 1), are 1, 1. 5, and 2 MPa respectively, normal values f or a pure-gum vul­canizate at different increasing degrees of cross-li nking.

The similarity of shape of the curves is obvious and suggests immediately a normalized plot in which the ordinate is F/M, the ratio of the stress to the modulus. Such a plot of course is identical with the lowest curve, (w here M = 1), if one utilizes the ordinate scale on the right.

M = 2.0 MPa

4 1.5 4

MPa 1.0 f/M

----+

10

FIGURE 1. Relation of stress F (based on undeformed cross-section) to extension ratio L.

ModulusM~l, 1.5, or 2 MPa. Lowest modulus curve is also normalized relation between FIM and L.

Thus the stress-strain relation for all three rubbers is represented by the single plot of F / M. Consequently attention can be concentrated on the shape of the curve without regard to variations in modulus. The modulus itself will vary with the nature of the rubber, the extent of vulcanization , the temper­ature, the time of creep or relaxation and possibly other fac tors. Howe ver , the relation of stress to strain does not appear to depend on what particular combination of these factors leads to a given modulus.

Martin, Roth, and Stiehler [2] developed the following empirical equation to represent the normalized relation up to L = 3.

F/M = (L-1 - L- 2) exp A (L - L- 1) (3.1)

where A is a parameter equal to about 0. 38. The lowest value they re ported was 0.32 and the highest 0.42.

The slope of the normalized equation , readily obtained by differentiation of eq 3.1 is

a(F/M)/aL = L-1 [exp A(L - L- 1)] [A(l (3.2 ) - L-1 + L- 2 - L-3 ) - L-1 + 2L-2].

With A = 0.38 there is a region of decreasing slope beginning at L = 0.26, passing through unit slope at L = 1 and having a point of inflection at L = 2.91. Beyond this the slope increases up to the point of rupture. The equation represents the previously-mentioned data of Martin, Roth, and Stiehler [2] for pure-gum rubbers reasonably well over the interval from L = 0.5 to about L = 3.5. As described more fully in the next section, in recent years other observers [14-17] have reported measurements on rubbers containing fillers, where the value of A was larger than 0.38, in some cases becoming as much as 1.1. Figure 2 is therefore drawn to represent the generalized normalized Martin , Roth and Stiehler equation where the parameter A can assume values other than 0.38. The lowest CUl-ve is the same as that shown in figure 1.

7.2

4

F/M

O~~~-L------L-____ -L ______ ~ ____ ~ o 10

FIGURE 2. Relation of normalized slress FIM (ratio of stress to modulus) to extension ratio L, as calculated by Martin, Roth, and Stiehler equation with coeffccient A = 0.38, 0 .50 , or 1.0.

58

The additional curves main tain unit slope at L = 1, but the point of infl ection is displaced downwards with increasing values of A. When A = 0.5 the point of inflection is at L = 1. 77. When A = 1.0 the slope obtained from eq (3.2) is unity within 0.6 pe rcent for 1.0 < L < 1.2. with an increasing slope at higher elongations.

4. Application of MRS Equation to Experimental Data

The validity of the representation of any stress-strain data by eq (3.1) can be ascertained by a plot of log F /(L - I - L - 2) against (L - L - I). If the data conform to the equation the plot will be linear. The modulus M can be obtained from the intercept (which is log M) , and the parameter A from the slope (which is 0.4343 A for base-lO logarithms). For brevity a graph with these co-ordinates will be called an MRS plot.

4.1. Pure-Gum Vulcanizates

Landel and Stedry [18] reported excell ent adhe rence to linearity in plots of this sort for samples of polyurethane . The data we re obtained at a constant strain rate (0.325 min- I in one case), rather than being isochronal as in the studi es of Martin, Roth , and Stiehle r [2] . The values of A, reasonably constant for a give n rubbe r, varied be tween 0.4 and 0.5. Linearity was obtained up to the breakin g strai n of 225 pe rcent. Similar s tudies with s tyrene-butadiene rubber (SBR) showed values of A varying from 0.35 to 0.45 at temperatures from - 5 to 60 °C for times from Ito 10,000 min. An average value of 0 .40 was utilized in later calculations. Both constant stra in rate and stress relaxat ion measurements were made with this rubber. At th e lowest temperature the measured strai n was as high as 550 pe rcent.

In later work Landel and F'edors [19] utilized the eq uation with a value A = 0. 40 to represent velY well the form of the stress-strai n relation for the fa ilure envelope of a mas ter curve with experimental data for 12 different rubbe rs.

T. L. Smith [20] stud ying the res ponse of pure gum SBR vulcaniza tes at widely different consta nt strain rates and temperatures obtained data which gave an eq uilibrium stress­strain curve represented by the MRS equation with A = 0.47 or 0.49 [21]. In late r work [22] the ultimate values of stress and strai n were also found to lie along the same curve .

Harwood and Schallamach [23, 24] made isochronal plots of data from constant-strain-ra te experiments for a pure-gum vulcan izate of natural rubber. They obtained good agreement with the s traight lines predicted by the MRS plot. At -45 o( the value of A was 0.56 for constant times from 0.2 to 1000 s. Over this time range the modulus M decreased from 41 to 13 kg/c m- 2 (4.0 to 1.27 MPa). The same value of A was obtained at other temperatures.

Bartenev and Vishnitskaya [25] report values of A and M dete rmined by MRS plots for several differe nt rubbe rs. With measurements in the ir own laboratories they report 0. 356 and 10.0 kg/cm2 (0.98 MPa) respectively, for a natural rubber vulcanizate as well as 0.371 and 7.24 kg/c m2 (0.710 MPa) for a styre ne-butadi ene rubber. In concluding that the equa­tion is fairl y sati s fac tOlY up to the point of rupture they also report values obta ined from MRS plots of data obtained by other observe rs [26-30].

Other s tress-strain data in the literature which we have found to show good approximations to linearity in an MRS

59

plot include observations by Shepard and Clapson [4], Tre­loar [6], Rivlin a nd Saunders [31]', Smith, Greene, and Ci fferi [32], and Obata et al. [33]. The MRS plots for the first three sets of observations have been presented in a previous paper [3].

J. F. Smith [34] has noted an abrupt increase in the slope of some MRS plots a t a high elongation. He ascribes this discontinuity of slope to the onset of crystallizat ion. From its location he and his co-workers [35] have drawn conclusions regarding the critical elongation as it is affected by diffe rent types of vulcanization. We have also found that some of the data presented by Smith , Greene and Cifferi [32] likewise show this a brupt increase in slope associated with crystalliza­tion.

4.2. Rubber-Filler Systems

Data obtained by Becke r and Rademac he r [15] on vulca n­ized rubbe r conta ining diffe rent amounts of titanium dioxide fill er a re suitable for this type of analysis. Va lues read from the ir figure 8 for the unfilled rubber are almost identi cal with those show n in the lowest curve of fi gure 1 of the present paper. Values for the fill ed rubbers lie above this curve.

Figure 3 presents a plot of Bec ker and Rademacher data (as read from the curves of the ir fi g. 8) in the form used to tes t the validity of eq (3. 1). The po ints are in excellent agreement wi th straight lines. The values of A inc reased from 0.38 for the pure gum rubbe r to 0. 50, 0.55, and 0.58 as the frac tional fill e r content was inc reased from 0 to 0. 1, 0.2, and 0. 3 res pectively. Values of the modulus M , as read from the inte rcept, inc reased fro m 1.03 MPa for the pure-gum rubbe r to 1.26, 1. 71, a nd 2.15 MPa respectively.

3.6

3.2

'.., ~ 1 I

N

~ 28 E .

15 8

2.0 "'-___ L-___ L-___ L-___ L-__ ---'

o 10 L- L- 1

FIGURE 3. MRS plot-ReLation of log F/(L - I - L - 2) to (L - L - I)for four voLume-fractions of Ti02 ji.Ller from. 0 to 0.3 in naturaL rubber.

Data of Becker and Ha demacher 11 5) (t he ir fi g. 8).

Similar results were obta ined in unpubli shed studi es by Nakayama [14] on rubber containing carbon black. He found somewhat higher values of A and some dev iations from linea r­ity at low elongations, with low values of A, for rubbers conta ining large amounts of fille r. The values of A obtained were 0.495, 0. 587, 0.732, and 0.951 for natura l rubber containing 0, 10 , 30, and 50 percent carbon blac k. Experi­mental details regarding his method of measurement and the

type of black are not available. Harwood and his co-workers [16, 17] have found linear MRS plots for isochronal data on styrene-butadiene rubber. The value of A for a pure-gum vulcanizate was 0.384 and did not vary with temperature (240 to 420 K). For an SBR vulcanizate containing 30 phr of HAF carbon black the values of A decreased from about 1.1 at 250 K to 0.83 at 425 K. Here, as with Nakayama's results, there were some deviations at low elongations. For each vulcani­zate, A was independent of the rate of extension over the range 0 . 1 to 1000 percent/so In a natural rubber containing mainly polysulfide cross-links the value of A was found to Increase, from 0.39 to 0.61 as the cross-linking was in­creased . .

5. Young's Modulus from Experimental Observations of Uniaxial Extension

and Compression

Young's Modulus M is defined as the limit of the ratio of stress to strain in the undeformed state (where L = 1). For the data considered here it will be equal to 3G where G is the shear modulus. We consider it highly desirable to reserve the word modulus for the limit of the ratio, not using it for the ratio itself or for F /(L - L - 2) as several previous workers have done.

In a previous paper [3] consideration was given to the various methods of obtaining M from observations of s tress and strain. The paper concluded that the most satisfactory method was to measure the intercept of an MRS plot (e. g., as in fig. 3), since a linear extrapolation from finite deformations to the unstressed state can include observations over a greater range than with other methods. Furthermore data on uniaxial compression can be included in the determination. This makes possible a determination of the modulus from the intercept obtained by interpolation of observed values of positive and negative uniaxial s tress rather than by extrapola­tion of positive stress values only.

A simpler form involving the plotting of F /(L - 1 - L -2) against L or the elongation (L - 1) was also found to be satisfactory (0.5%) when L is between 0 .5 and 2. O.

Previous papers [36, 37] have already called attention to the fact that the modulus as obtained by the use of the MRS equation in uniaxial extension is in excellent agreement with the modulus obtained directly by an indentation technique with specimens cross-linked by varying amounts of dicumyl peroxide [38]. Good agreement was also obtained with the modulus on similar specimens measured by a torsion pendu­lum [39].

6. Mooney-Rivlin Equation

Two different independent types of theoretical approach by Mooney [40] and by Rivlin [41] lead to an equation which may be written

(6.1)

where S 1 and 52 are, in general, functions of L. It is of interest to examine experimental stress-strain data to see whether there are certain regions where 5 1 and 52 are found to have constant values.

The usual procedure to locate these regions is to put the equation into the form

60

(6.2)

In those regions where S 1 and 52 are constant, a plot of the left-hand member F /[2(L - L -2)] against L -1 will yield a straight line with an intercept 51 and a slope 52' Let us denote these constant values as eland C 2 respectively and call a graph with these coordinates a Mooney-Rivlin plot.

Figure 4 shows such a plot for the stress-strain data for the typical pure-gum vulcanizate exemplified by the middle curve offigure 1 , where M = 1.5 MPa. It can be seen that in extension there is indeed a region of linearity where the curve coincides with the dashed line shown. This region includes values of L- 1 between 0.3 and 0.7 , namely where L is between about 1. 5 and 3.5. However there is increasing divergence between the curve and the straight line on ap­proach to the undeformed state, where L -1 = 1.

::< -B-

L

360.-~5~4~3~-T2 __ ~1 .~5 ____ ~1 ~.0 __ ~~ ______ ~~ __ ~0.5

k Po

300

" 200

~ I~ 100

/

/ /

/

(,

A ~ 0.38

OL-________ L-________ L-________ L-______ ~ o 0.5 1.0 1.5 2.0

l-'

FIGURE 4. Mooney-RivLin pLot -ReLation oj F/[2(L - L - 2)] to L - 1 Jar rubber conJorming to MRS equation with A = 0.38.

The curve represents experimental data, which conform to eq (3.1), the MRS equation . In terms of the ordinate offigure 4 the equation is:

F M(L-I - L- 2 ) ~~- = expACL - L- l) (63) 2(L - L -Z) 2(L - L -2) .

The equation of the straight line is

(6.4)

The straight line represents experimental data only in the region where it coincides with the curve .

l

It is ev ident from figure 4 that

F M C I < lim ( 2\ = - < (el + C2 )· (6.5)

L=I 2 L - L - ) 6

The shea r modulus G = 1/3 M is often calculated by cross­linking cons ide rations [42 , 43]. It is clear that it should not be regarded as directly correlated with either C I [44] or with (C I + C2) [45]. The value of M16, always intermediate between the two , can best be determined by the method given in Section 5. The Mooney-Rivlin equation with constant coeffi cients certainly does not agree with the experimental data (represented by the curve) in the region where L - I is greate r than 0.7 - namely where L is less than 1. 5.

AtL = 1 the ordinate of the curve , namely the ratioFI[2(L L - 2)], has both numerator and denominator ze ro, but its

value may be obtained by diffe renti ating them both and applying the rule of de I'HopitaJ. The ratio of the derivatives gives the ordinate of fi gure 4 at L = 1 as M 16 where M is [aF I aL]L=J' The actual value is 250 kPa in thi s case .

The values of C I and C2 may be read from fi gure 4 . Subjective variations in drawing the line may be avo ided by us ing the analytical express ion of the MRS equation with A = 0.38 to represent the stress F. As outlined in more detail previously [3] the analys is shows tha t the re is a point of infl ection in the curve nea r L - I = 0.42 and a minimum at L- I

= 0.24. If the stra ight line is drawn as the tangent at the point of infl ection , it will be found tha t C1 = 0 .64 M = 96 kPa and C2 = 0 . 124 M = 186 kPa , with the ratio C2/C J = 1. 94. Thus the ordinate of the straight line at L = 1, namely (C I + C2) is 282 kPa. As expected this is definitely greate r than 250 kPa , the ordinate of the curve at thi s point. If the straight line is drawn to be tangent to the curve at a point where the abscissa L - I is greater than 0.42 , somewhat lower values of C2 will be obta ined, but poorer agreement is found below 0.42.

Many previous autho rs have assumed the constancy of the Mooney-Rivlin coe ffi c ients 5 J = Chand 52 = C2 in the region where L - I is be tween 0.7 and 1, (1.0 < L < 1.5). The fallacy in this assumption, (at least for a rubber with A = 0.38) is shown by an examination of fi gure 4, where the curve representing experimental values deviates from the dotted line representing the Mooney-Rivlin equations.

In the compression region, where L- J > 1, there is general agreement that experimental data do not conform to eq (6.2) with the values of C 1 and C 2 applicable in the extension region. Instead the value of F 1[2(L - L -2)] is approximately constant with a value of about (1/6)M as predicted by the statistical theory of rubber elastici ty. In terms of the Mooney­Rivlin equation, C tiM = 0. 167 and C21M = 0 in this region.

The value of F 1[2(L - L - 2)] predicted by the MRS equa­tion, eq (3. 1), with A = 0.38 shows only a small variation between L - I = 1 and L - ] = 2 as can be noted in figure 4. In the normalized form the calculated value of F 1M in this region differs from that calculated from the stati sti cal theory, namely 1/3 (L - L - 2), by less than 4 percent [3]. Since the prec is ion of the available experimental data in this region is no better than this fi gure, the MRS equation may be regarded as an adequate representation of the data for compressions as large as those in which the thickness is reduced to half the original thickness (L = 0.5).

A normalized Mooney-Rivlin plot is shown in figure 5 where the ordinate of figure 4 has been divided by the

L

.19 r-__ 3:;---:;.._1c,:.5'----.--'1:;.0 ___ -'.::.6.:r67'--__ ---'.::;.5 ___ ---..:.:.4

.1 6

.15

4> .14

/ /

.13 / ~ ~.40

.12

.11

1.0 1.5

0 .38 '-.... 0 .40

2.0 2.5

FICURE 5. Normalized Mooney-Rivlin plot - Relation oj F/[2M(L - L- 2)1 to L - 1 Jor rubber .. conforming to MRS eqllation with A = 0 .36, 0.38, 0.40, 0 .50, 0.55, 0.60, or 1.0.

modulus M to give the curve marked A = 0_ 38. This curve can be utilized for a ll values of modulus including those whic h differ from the value (1.5 MPa) represented in fi gure 4.

Figure 5 also shows normalized Mooney-Rivlin plots for stress-strain data represented by seve ral different arbi trary va lues of A in the generalized MRS equation. It can be seen that C tiM increases and C 21M dec reases as A is rai sed above 0.38, with C21M becoming zero when A is near 0.55. C21M is nega tive when A is still hi gher. The point of infl ection and the sum (C J + C2) both show littl e change with inc rease of A, but the minimum ("u pturn") moves upward. In all cases the ordinate of the curve at L = 1 is 1/ 6 , whil e the ordinate of the straight line at this point is (C J + C 2)IM.

In the compress ion region of fi gure 5 the near-constancy of ordinate is soon lost as the value of A is increased above 0.4. For e las tomers ex hibiting this type of stress-strain behav ior, the statistical theory of elastici ty would not be eve n approxi­mately valid in compression. However, this conclusion has not been checked experimentally by a study of compression data on rubbers conta ining a filler.

7. Previous Use of Mooney-Rivlin Plots

During the past quarter century expel'imental values of stress and strain have been presented in the form of Mooney­Rivlin plots for hundreds of different elastomer systems. The conclusions of the present paper up to this point have been applicable to only those systems which conform to the gener­alized Martin , Roth and Stiehler equa tion , where A and M may have any values. The conformity in any particular case can be determined by the linearity of an MRS plot like fi gure 3. Mention has already been made in Section 4 of many cases in which this conformity has been found.

It will be remembered that in the original formulation the MRS equation was stated to be applicable to isochronal data - that each point of a given curve was obtained at the same time after the application of the stress. Some of the la ter work has found the relation valid also at constant rates of strain. In other work the stress has been measured after a

61

specified program of prestretching - often a program sug­gested by Gumbrell, Mullins, and Rivlin [46]. It is not certain to what extent these deviations from isochronal data may cause deviations from linearity in the MRS plots and resulting invalidation of the conclusions just drawn regarding the Mooney-Rivlin plots. '

With the foregoing qualification one can make several general statements about the literature results, many of which have been summarized by Blokland [10]. For example, in many cases the upper limit of linearity lies above L - I = 0.7. Sometimes it is as high as 0.9. However only rarely do the observations themselves extend above 0.85. In a number of instances the experimental points fall below the straight line for the highest values, as would be predicted from figure 4. In all the literature we have found at the higher values an almost complete absence of points lying above the line. Equal numbers above and below would be expected if the deviation arose from random errors of measurement of strain - the cause often assigned to this deviation [47, 48].

Figure 4 shows the lower limit of linearity of the Mooney­Rivlin plot as about L - 1 = 0.3 with a minimum at 0.24. In the literature many different values can be found but there is always an upturn immediately below the linear region, if the observations extend to high enough elongations. Mullins [44] found values of the upturn region as high as 0.6. The higher values are obtained as the cross-linking is increased or when measurements are made on a swollen specimen. He ascribes the upturn to an approach to the limited extensibility of a network, in contrast with some others [49] who consider it to be due to crystallization. Smith, Greene, and Cifferi [32] consider it as arising initially from the first and then from the second cause. It can be seen from figure 5 that increasing values of A, which might be produced by the increase of filler content or by crystallization, cause the upturn to appear at higher values of L -I.

In some instances metnioned in section 3, J. F. Smith [34, 35] found the onset of crystallization giving rise to an abrupt increase of slope of the MRS plot corresponding to an in­crease in A at high elongations. In these instances the upturn in the Mooney-Rivlin plot would be still more pronounced than that in figure 4.

The ratio C 2/C I is often found to be less than 1. 94, the value obtained when MRS coefficient A = 0.38. In many instances the ratio is found to be near 1. The ratio is a very sensitive function of A , as can be seen in figure 5. The ratio is expected to be near zero when A is near 0.55 as already noted. The ratio often depends on molecular weight of the rubber and on previous thermal and mechanical history.

In view of the inability of previous workers to relate the Mooney-Rivlin constants C I and C2 to molecular considera­tions and the failure of the equation to represent experimental data in the region of low elongations or compression as shown here, we think that Mooney-Rivlin plots have only very limited utility. It seems futile to try to ascribe much signifi­cance to C 2 after noting the wide variations shown in figure 5.

Jones and Treloar [47] have recently shown that C 1 and C 2

do not furnish significant information about the strain energy, and that (contrary to Mooney's assumption [40]) the shear stress-strain curve is not linear. They show a curve very similar to figure 4 with a maximum near L -I = 2 and extending smoothly into the region of low extensions. Their experimental data are represented by a stress-strain relation much more complex than the generalized MRS equation.

62

8. Conclusions

A comprehensive literature survey shows the general ap­plicability of the generalized normalized Martin, Roth, and Stiehler equation to uniaxial stress-strain data in extension and compression on rubber vulcanizates. The equation can be expressed as F 1M = (L - I - L -2) expA (L - L - 1) where F is the stress on the undeformed section and L the ratio of stressed to unstressed length. The equation contains two constants - M, Young's Modulus, the slope of the stress­strain curve at L = 1, and A an empirical constant.

The conformity of stress-strain data to the equation can readily be determined by a plot of log F /(L - I - L - 2) against (L - L -I).

In almost every case a straight line is obtained, from the slope and intercept of which both the constants can be determined. The range of validity of the equation usually begins nearL = 0.5 (in the compression region) and continu­ing through the region of low deformations often extends to the region of rupture in extension. If uniaxial compression data are available the modulus can thus be obtained by interpolation through the region of low deformations, where experimental data are often somewhat unreliable. The value of the modulus M varies with the nature of the rubber, the extent of vulcanization, and the time and temperature of creep or stress relaxation. The value of the constantA is near 0.4 for pure-gum vulcanizates, increasing to values near 1.0 with increasing filler content, and showing an abrupt increase when crystallization occurs. Direct experimental observations where the deformation of a single specimen is varied continu­ously from compressive to tensile deformation are cited to show that M, defined as the limit of the ratio of stress to strain, is independent of the direction of approach to the limit at L = 1.

The normalized Mooney-Rivlin plots show F 1[2M(L -L - 2)] against L - I. These graphs have only limited regions of linearity corresponding to constant values of the coefficients C I and C2 • Since these regions do not include the undeformed state the Mooney-Rivlin equation can not be used at low elongations or in compression. The values of C I and C 2 show very wide fluctuations for the Mooney-Rivlin plots of experi­mental data, which are themselves usually well represented by the Martin, Roth, and Stiehler equation with different values of the constant A.

In view of all these considerations the conclusion of the present study confirms that of Treloar in his recent publica­tions [7, 50] in failing to find much utility in making Mooney­Rivlin plots. The failure to represent the experimental data at low elongations and the inability to correlate the constants with theoretical predictions based on strain energy or statisti­cal theory considerations are the most serious objections.

9. References

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